Question

A dental hygiene company is creating a new 24 -ounce tube of toothpaste by combining its most popular toothpastes, Cavity Crusher and Bad Breath Obliterator. Cavity Crusher contains $0.25\mathrm{\%}$ of sodium fluoride as its active ingredient, and Bad Breath Obliterator contains $0.30\mathrm{\%}$ of triclosan as its active ingredient for a total of $0.069$ ounces of active ingredients in both toothpastes. Solving which of the following systems of equations yields the number of ounces of Cavity Crusher, $c$, and the number of ounces of Bad Breath Obliterator, $b$, that are in the new toothpaste? A) $$\begin{array}{rl}c+b& =0.069\\ 0.25c+0.3b& =24\end{array}$$ $$\begin{array}{rl}c+b& =24\\ 0.0025c+0.003b& =0.069\end{array}$$ B) C) D) $$\begin{array}{rl}c+b& =24\\ 0.025c+0.03b& =0.069\\ c+b& =24\\ 0.25c+0.3b& =0.069\\ \frac{2d^2-d-10}{d^2+7d+10}& =\frac{d^2-4d+3}{d^2+2d-15}\end{array}$$

Short answer

The correct system of equations is: B) $$\begin{array}{rl}c+b& =24\\ 0.0025c+0.003b& =0.069\end{array}$$

Step-by-step solution

Write down the given information

We know that there is a total of 24 ounces in the new toothpaste tube and a total of 0.069 ounces of active ingredients. We also know that Cavity Crusher has a 0.25% contribution of sodium fluoride and Bad Breath Obliterator has a 0.30% contribution of triclosan. Let c be the number of ounces of Cavity Crusher and b be the number of ounces of Bad Breath Obliterator.

Write the first equation from the total ounces information

We know that the total amount of toothpaste in the new tube will be the sum of the ounces of Cavity Crusher and Bad Breath Obliterator. Therefore, our first equation can be written as: $c+b=24$

Write the second equation from the active ingredient information

Knowing that there are 0.069 ounces of total active ingredients, we can say that 0.25% of sodium fluoride from Cavity Crusher and 0.30% of triclosan from Bad Breath Obliterator contribute to the total active ingredients. To find the amounts of active ingredients from each toothpaste, we need to use the given percentages. The equation can be written as: $0.0025c+0.003b=0.069$ Now that we have both equations, we can identify the correct system of equations:

Compare with the listed systems of equations and pick the correct one

We have found our system of equations to be: 1) $c+b=24$ 2) $0.0025c+0.003b=0.069$ Comparing these equations to the choices in the problem, we find that the correct choice is: B) $$\begin{array}{rl}c+b& =24\\ 0.0025c+0.003b& =0.069\end{array}$$

Key concepts

Systems of Equations

When confronting SAT math problems, one of the common topics you will encounter are systems of equations. A system of equations is a set of two or more equations that you deal with all at once. Each equation has two or more variables, and the key is to find the values of these variables that satisfy all equations simultaneously.

In the context of SAT problems or other real-life situations like mixing ingredients for a product, systems of equations can be used to model relationships between different quantities. Effective strategies to solve these systems include substitution, elimination, graphing, and using matrices.

For instance, with the problem provided, we use the system of equations to find how many ounces of each toothpaste—Cavity Crusher, represented by 'c', and Bad Breath Obliterator, represented by 'b'—make up the new product. The first equation stems from the total ounces 'c + b = 24' and the second reflects the active ingredients '0.0025c + 0.003b = 0.069'. We solve this specific system by comparing the equations we've formed with the given options and identifying the matching system.

Percentages in Math

Working with percentages in math is a fundamental skill, especially on the SAT. Percentage problems can include finding the part given the whole and the percentage, finding the whole given the part and the percentage, or finding the percentage given the part and the whole. In the dental hygiene company's problem, percentages denote the concentration of active ingredients in each toothpaste type.

It’s essential to be comfortable converting percentages into decimals for computation, as was necessary in the problem. To convert a percentage to a decimal, divide it by 100. For example, in our exercise, '0.25%' is converted to a decimal to be used in the equation leading to '0.0025'. This allows us to write an equation representing how much of each active ingredient is in the toothpaste mixture. Students should practice these conversions, as percentage-related questions are very common on the SAT.

SAT Test Preparation

SAT test preparation is key to performing well on the exam. The SAT Math section, which often includes questions on systems of equations and percentages, requires a strong understanding of fundamental concepts and the ability to apply them in various contexts. To prepare effectively:

• Review key concepts such as algebra, geometry, and data analysis.
• Practice with a variety of problems to become versatile in solving different types of equations.
• Work on time management, a critical skill given the SAT's time constraints.
• Take practice tests to familiarize yourself with the exam's format and question types.
• Understand and learn from your mistakes to improve your problem-solving strategies.

Preparing for the SAT is not just about knowing math; it's also about honing your test-taking techniques and managing your exam time strategically to maximize your score.